Usually driven and ambitious, this character aims to constantly hone his skills by tackling challenges that push his abilities to the limit, and enjoys every minute of it. The harder the challenge, the better; they may even become bored with easy tasks or disappointed if something turns out to be easier than they expected.

Sister Trope of Spirited Competitor, but this type of character doesn't necessarily need a Worthy Opponent to compete against. It's more about the game/sport than who they play against, although they will usually be happy to have a worthy friendly rival who can challenge them and test their skills. Shōnen anime protagonists in particular tend to exhibit this trope, especially those who "wanna be the very best."

Compare the Blood Knight, for those who specifically like a good fight.

Examples:

Anime and Manga

Endou in Inazuma Eleven. On more than one occasion, he actually got excited from seeing the opposing team score a goal on him or otherwise getting his butt kicked, because it meant he had a chance to play against amazing opponents. Hiroto also shows bits of this mixed with Spirited Competitor when he seeks out Raimon (and Endou in particular) for a match.

This is a lot of the stronger characters in the Pokémon series. Cynthia in particular is now at the point where she automatically identifies determined-looking ten-year-olds as godly powerful, which for her is reason enough to single them out for a challenge.

From The World God Only Knows, main character Katsuragi Keima. He plays multiple games at once to mostly save time, but partly to challenge himself.

One of the defining features of Goku in Dragon Ball. He even rejects sensible solutions for tackling enemies (e.g. shutting down the androids) in favor of taking on strong opponents face-to-face.

Saiyans in general tend to be this and Vegeta is just as bad if not worse than Goku in this aspect. He allowed 17 and 18 to be re-activated because he found the fight with Androids 19 and 20 to be boring and he also allowed Cell to reach perfection because he was bored by how easily he was beating Cell's semi-perfect form.

Comics

In Marvel Comics the Champion of the Universe goes around to various worlds looking for a good boxing match. His first defeat ever comes from The Thing, who refuses to stay down. He can beat the Thing but he can't defeat him, so the Champion concedes.

The Grandmaster has spent his extraordinary long life in pursuit of amusement through the playing of games. He has traveled throughout the known universe studying many civilizations' forms of games and play to the point of mastering them. He then began to devise his own types of tournaments and contests, challenging various opponents to games of skill and chance for high stakes.

Fan Works

Darius Walker from Honorable Hogwarts has shown a strong tendency toward this. He's fine using his wand against lesser opponents (or just poisoning them and sparing himself the tedium of a quick and easy fight), but when he heard what a good swordsman Jason was he came after him with a blade instead, to test his skills against the best.

In Tin Cup, Roy destroyed his chance at becoming a pro golfer when he was younger because he had a tendency to forego safe shots in critical moments for impossible ones just because he prided himself on taking risks that others won't. He also can play golf with unconventional clubs (like shovels and pool cues) and loves to show off his skill at trick shots.

Soap: Jessica's attorney E. Ronald Mallu takes her case (she's on trial for Peter Campbell's murder) because it's so difficult. If he loses, it won't harm his reputation because nobody expects him to win, and if he wins it'll be a feather in his cap. Another attorney had said that he couldn't get Jessica acquitted with a mute prosecutor, her husband as the judge, and her family as the jury.

Wulfrik the Wanderer of Warhammer once drunkenly boasted that he could defeat any foe in this world or any other. When he awoke the next morning, he found himself able to speak any language and cursed to hunt down all possible challengers (the gift of tongues was so that he could issue a challenge they could not refuse). He's greatly enjoyed himself ever since.

Vyse fits certainly the bill. In Skies of Arcadia, he repeatedly tackles challenges others deem impossible: such as escaping the Grand Fortress and crossing the Dark Rift. He even has a special ability called "challenge lover" in Valkyria Chronicles. Vyse is not only made of this Trope, he eats it for breakfast.

Vyse (to Marco): "Impossible" is a word people use as an excuse when they want to give up. Whenever someone says something's impossible, it makes me want to prove them wrong.

Othello was born into a family of means, and never had to work to support himself. Instead, he challenged himself with the most deadly jobs and in adrenaline-fueling adventures around the galaxy - climbing the ice floes of Mt. Karanish, participation in the Liandri Grand Tournament, even mercenary work in against-all-odds military conflicts.

Zer0 of Borderlands 2 seeks a challenge above all else, with his personal ECHO entry telling one of his targets to make him work for it. He becomes a Vault Hunter because he sees it as the ultimate challenge. Unfortunately, he sometimes takes this a bit too far as he'll assume that the "challenge" of shutting up an idiot means that he can chop someone's head off. Axton and Salvador have similar motivations, though they also really want the fame and loot (respectively).

Levi the Slasher in Magical Girl Lyrical Nanoha As Portable: The Gears of Destiny. Being the Material of Power, she loves nothing more than having strong enemies to test her mettle against. At the end of the game, we see her drooling at the chance of going to Eltria after she learns that it's essentially a Death World filled with powerful monsters to fight and mysterious dungeons to explore. She then proceeds to unintentionally save a lot of villages in Eltria, as she spends her time stopping several monsters threats for her own enjoyment.

One of the reasons that Sly Cooper and his gang steal from criminals is that they believe that stealing from ordinary people is boring and the true challenge comes from beating criminals at their own game.

Western Animation

The Dial M for Monkey episode "Rasslor". The title character comes to Earth and challenges its superheroes to wrestle him. He says, "For eons I have scoured the cosmos searching for the one adversary who could provide me with suitable sport." If none of the heroes can defeat him, he will destroy the planet. All of the superheroes go down to defeat except Monkey, who refuses to give up even after Rasslor beats him up.

Rasslor: I could crush your body, I could smash your bones, but I could never break your spirit. You are a marvel, little monkey! Any world that could spawn one as noble as you is truly blessed. I spare your Earth!

Stinkoman from Homestar Runner. "Are you asking for a CHALLEEEEEEEEEEENNNNNNNNNNNGGGGGGGGGEEEEEEEEEEEEE?!!!"

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